Let $M$ be a complete, connected Riemannian manifold without boundary. Given a point $p\in M$ and a subset $K$ of $S_p M$, the unit sphere in $T_p M$, define the $K$-cone of directions $C(K)$ around by $C(K) := \{w \in T_p M| \ w/|w| \in K\}$.

Given $p$ and $K$ as above, define $\text{inj}(p, K)$ to be the supremum of values $r \geq 0$ for which the exponential map at $p$ is injective on $C(K)$ intersect $B_r (0)$.

Suppose that

$\sup_{p \in M} sup_{K \subset S_p M, K \text{ non-null}} \text{inj}(p, K) < \infty$.

Does it follow that the manifold $M$ is compact?

Note: By non-null, I mean not of null measure as a subset of the smooth manifold $S_p M$.